Method of infinite system of equations for problems in unbounded domains Dang Quang A Institute of Information Technology 18 Hoang Quoc Viet, Cau giay, Hanoi, Vietnam Innovative Time Integration, May 13-16, 2012 Innsbruck, Austria May 16, 2012
Outline 1 2 Infinite system of equations Quasi-uniform grid 3 4 5
Many problems of mechanics and physics are posed in unbounded domains: heat transport problems in infinite or semi-infinite bar, aerosol propagation in atmosphere, problem of ocean pollution,... For solving these problems one usually restricts oneself to treat the problem in a bounded domain and try to use available efficient methods for finding exact or approximate solution in the restricted domain.
Questions: which size of restricted domain is enough and how to set conditions on artificial boundary for obtaining approximate solution with good accuracy?. 1 Simplest way: to transfer boundary condition on infinity to the artificial boundary. 2 Set appropriate conditions on artificial boundary (Dang, Tsynkov, Colonius,Halpern,... ). 3 Set transparent conditions on artificial boundary ( Arnold, Ehrhardt, Schulte, Sofronov,...)
Quasi-uniform grid for mapping unbounded domain to bounded one ( Alshin, Alshina, Boltnev, Kalitkin). We approach to problems in unbounded domain by infinite system of linear equations.: construct difference scheme for the problem in unbounded domain suggest a method for treating the infinite system in order to obtain an approximate solution with a given accuracy. This infinite system approach overcomes drawbacks of the quasi-uniform grid method.
Infinite system of equations Quasi-uniform grid Infinite system of equations (Kantorovich and Krylov) Canonical form: x i = c ik x k + b i, i = 1, 2,... (1) k=1 The solution found by the method of successive approximation x (n+1) i = k=1 c ik x (n) k + b i, i = 1, 2,...; n = 1, 2,... with the zero starting approximation x (0) i = 0 (i = 1, 2,...) is called the main solution of the system.
Infinite system of equations Quasi-uniform grid Set ρ i = 1 c ik. k=1 The system (1) is called regular system if ρ i > 0, i = 1, 2,...; it is called completely regular if there exists a constant θ > 0 such that ρ i θ, i = 1, 2,... Assume that there exists a number K such that the free members b i satisfy the condition b i Kρ i, i = 1, 2,... (2)
Infinite system of equations Quasi-uniform grid Theorem The regular system (1) with the free members satisfying the condition (2) has a bounded solution x i K which can be found by the method of successive approximation.
Infinite system of equations Quasi-uniform grid Theorem The main solution of the regular system (1) with the free members satisfying the condition (2) can be found by the truncation method, that is, if xi N is the solution of the finite system N x i = c ik x k + b i, k=1 i = 1, 2,..., N then x i = lim N x N i, where xi (i=1, 2,...) is the main solution of the system.
Infinite system of equations Quasi-uniform grid Quasi-uniform grid (Alshin, Alshina, Kalitkin and Panchenko, 2002) Let x(ξ) be strictly monotone smooth function of the argument ξ [0, 1]. The grid ω N = {x i = x(i/n), i = 0, 1,..., N} with x(0) = a, x(1) = + is called quasi-uniform grid on [a, + ]. In this case the last node x N of the grid is on the infinity. Example of quasi-uniform grids are the grids ω N = {x i = i, i = 0, 1,..., N} (hyperbolic grid), N i ω N = {x i = tan πi, i = 0, 1,..., N} (tangential grid). 2N
The model problem of heat conductivity in a semi-infinite bar (ku ) + du = f (x), x > 0, u(0) = µ 0, u(+ ) = 0 (3) Difference scheme on the uniform grid {x i = ih, i = 0, 1,...} 1 ( ) y i+1 y i y i y i 1 a i+1 a i + d i y i = f i, i = 1, 2,... h h h y 0 = µ 0, y i 0, i, a i = k(x i h/2), d i = d(x i ), f i = f (x i ).
Canonical form of infinite system y i = p i y i 1 + q i y i+1 + r i, i = 0, 1, 2,... y i 0, i. (4) Progonka method for tridiagonal system of equations: y i = α i+1 y i+1 + β i+1, i = 0, 1,..., (5) where coefficients are calculated by the formulas: α 1 = 0, β 1 = µ 0, q i α i+1 =, β i+1 = r i + p i β i, i = 1, 2,... 1 p i α i 1 p i α i (6)
Theorem Given an accuracy ε > 0. If starting from a natural number N there holds β i 1 α i ε, i N + 1 (7) then for the deviation of the solution of the truncated system ȳ i = p i ȳ i 1 + q i ȳ i+1 + r i, i = 0, 1,..., N, ȳ i = 0, i N + 1 (8) compared with the solution of the infinite system (4) there holds the following estimate sup y i ȳ i ε. (9) i
Example ((1 + 1 1 + x )u ) + (1 + sin 2 x)u = f (x) u(0) = 1, u(+ ) = 0, Exact solution u(x) = 1 1+x 2. Table: Case h = 0.1 ε N error 0.01 108 0.0085 0.001 325 0.0012
Exact solution u t = k 2 u, 0 < x < +, t > 0, x 2 u(x, 0) = 0, u(0, t) = 1, u(+, t) = 0. u(x, t) = 2 π + x/2 kt exp( ξ 2 )dξ. (10) Infinite system on each time layer j + 1 : where r = kτ/h 2 ry j+1 j+1 i 1 + (1 + 2r)yi ry j+1 i+1 = y j i, i = 1, 2,... y j+1 0 = 1, y j+1 i 0, i, (11)
Figure: By infinite system Figure: By quasi-uniform grid
Stationary problem of air pollution u ϕ x w ϕ g z x ν ϕ + σϕ = 0, x > 0 z uϕ = Qδ(z H), x = 0 ϕ = αϕ, z = 0, ϕ 0, z, z The numerical solution on uniform grid using the infinite system method was studied by Dang (1994), where a theorem similar to the above Theorem was proved.
Problem describing ion wave in stratified incompressible fluid 2 ( 2 ) u t 2 x 2 u + 2 u = 0, x > 0, t > 0, x 2 u(0, t) = f (t), u(+, t) = 0, u u(x, 0) = f 1 (x), (x, 0) = 0. t Set φ = 2 u x 2 u. Then the problem is decomposed into { 2 φ t 2 + φ + u = 0, φ(x, 0) = f 1 (x) f 1(x), φ t (x, 0) = 0, 2 u u = φ(x, t), x 2 u(0, t) = f (t), u(+, t) = 0.
Difference schemes φ j+1 i 2φ j i + φj 1 i τ 2 + φ j i + uj i = 0, j = 1, 2,... φ 0 i = f 1 (x i ) f 1 (x i ), φ 1 i = φ 0 i. uj+1 i 1 2uj+1 i + u j+1 i+1 h 2 + u j+1 i = φ j+1 u j+1 0 = f j+1, u j+1 i 0, i. i, (12) (13)
Example 1: Take initial conditions be homogeneous and the left boundary condition u(0, t) = arctan 2 (10t). sin(0.3t). Figure: By infinite system Figure: By quasi-uniform grid
Example 2: The left boundary condition is zero, the initial condition is u(x, 0) = x 3 e x, u (x, 0) = 0. t Figure: By infinite system Figure: By quasi-uniform grid
Summary and outlook Propose and investigate the infinite system method for solving several one-dimensional stationary and nonstationary problems, where the keystone is when truncate the infinite system. This method reveals advantage over the quasi-uniform grid method in time-dependent problems, especially in problems of wave propagation. In combination with the alternating directions method the method can be applied to two-dimensional problems in semi-infinite and infinite strips?
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